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 locally_finite_topology_subset [2016/09/14 15:37]nikolaj locally_finite_topology_subset [2016/09/15 01:42]nikolaj Both sides previous revision Previous revision 2016/09/15 01:42 nikolaj 2016/09/14 15:52 nikolaj 2016/09/14 15:37 nikolaj 2016/09/14 15:34 nikolaj 2016/09/14 15:31 nikolaj 2016/09/14 15:30 nikolaj 2016/09/14 15:30 nikolaj 2016/09/14 15:28 nikolaj 2016/09/14 15:16 nikolaj created Next revision Previous revision 2016/09/15 01:42 nikolaj 2016/09/14 15:52 nikolaj 2016/09/14 15:37 nikolaj 2016/09/14 15:34 nikolaj 2016/09/14 15:31 nikolaj 2016/09/14 15:30 nikolaj 2016/09/14 15:30 nikolaj 2016/09/14 15:28 nikolaj 2016/09/14 15:16 nikolaj created Line 13: Line 13: Like many properties, this is a notion of smallness. It's not about the smallness of a subset $U$ of $X$, but smallness of a collection ${\mathcal C}$ of subsets $U$ of $X$. Like many properties, this is a notion of smallness. It's not about the smallness of a subset $U$ of $X$, but smallness of a collection ${\mathcal C}$ of subsets $U$ of $X$. - You may consider a well choosen sample of neighborhoods (the sets $V\in{\mathcal T}$) and ${\mathcal C}$ ought to be finite with respect to that sample (countable ​//pro// $V$). + You may consider a well choosen sample of neighborhoods (the sets $V\in{\mathcal T}$) and ${\mathcal C}$ ought to be finite with respect to that sample (finite ​//pro// $V$). === Dicussion === === Dicussion === - A topologal space is paracompact if it has a cover with that property. + * A topologal space is //paracompact// if it any cover has a refinement ​with that property. + * The sample of $V$'s above may be very big, so ${\mathcal C}$ is really only small w.r.t. the sample. In a //compact// space, on the other hand, the cover itself is finite (and you don't need to consider that sample). + * Note that the name //locally compact// is already used for the situation where every point $x\in X$ has a compact neighborhood $V$. === Reference === === Reference ===